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Mirrors > Home > ILE Home > Th. List > sucelon | GIF version |
Description: The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
Ref | Expression |
---|---|
sucelon | ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceloni 4417 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | eloni 4297 | . . 3 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴) | |
3 | elex 2697 | . . . . 5 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ V) | |
4 | sucexb 4413 | . . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 133 | . . . 4 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ V) |
6 | elong 4295 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
7 | ordsucg 4418 | . . . . 5 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) | |
8 | 6, 7 | bitrd 187 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord suc 𝐴)) |
9 | 5, 8 | syl 14 | . . 3 ⊢ (suc 𝐴 ∈ On → (𝐴 ∈ On ↔ Ord suc 𝐴)) |
10 | 2, 9 | mpbird 166 | . 2 ⊢ (suc 𝐴 ∈ On → 𝐴 ∈ On) |
11 | 1, 10 | impbii 125 | 1 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 Vcvv 2686 Ord word 4284 Oncon0 4285 suc csuc 4287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: onsucmin 4423 onsucuni2 4479 |
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